Array Codes for Functional PIR and Batch Codes

A functional PIR array code is a coding scheme which encodes some s information bits into a t\times m array such that every linear combination of the s information bits has k mutually disjoint recovering sets. Every recovering set consists of some of the array's columns while it is allowed to read at most \ell encoded bits from every column in order to receive the requested linear combination of the information bits. Functional batch array codes impose a stronger property where every multiset request of k linear combinations has k mutually disjoint recovering sets. Locality functional array codes demand that the size of every recovering set is restrained to be at most r. Given the values of s, k, t, \ell,r , the goal of this paper is to study the optimal value of the number of columns m such that these codes exist. Several lower bounds are presented as well as explicit constructions for several of these parameters.